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Confusion about a question, I am probably missing something, i.e. I'm not sure what the point of the question is.
Let $f_n$ be sequence of $\mu$ almost everywhere finite $\mathbb R$ valued functions, where $\mu$ is a $\sigma$-finite measure. Prove that there exists constants $c_n>0$ such that $\sum_n c_n f_n$ converges for almost every $x$.
Now, because the functions are almost everywhere finite, the only way this will converge is if the $c_n$'s tend to zero as $n$ goes to infinity. I'm confused how to use the measure here, should I be looking at subsets of $X$ and see if the result holds there?